UMVUE of geometric sample

Hi,

I have a an iid sample from a geometric (p) distribution. I found that the complete sufficient statistic is using "full exponential family with open set."

I am trying to find the UMVUE of .

I am arguing that the expectation of equals the expectation of my complete sufficient statistic and thus I use it as my estimator and it is unbiased. Furthermore, because the unbiased estimator is also complete, I am guaranteed that it is UMVUE.